BOUNDS FOR THE POSITIVE EIGENVECTORS OF NONNEGATIVE MATRICES AND FOR THEIR APPROXIMATIONS BY DECOMPOSITION

This paper deals with the positive eigenvectors of nonnegative irreducible matrices that are merely characterized by a given upper bound lambda on their spectral radius and by a given matrix L of lower bounds for their elements. For any such matrix, the normalized positive left (right) eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows (columns) of the matrix ( lambda I-L)** minus **1. This polyhedron is proved to be also the smallest closed convex set that is guaranteed to contain the positive left (right) normalized eigenvector; its vertices are therefore the best bounds one can obtain. These results are then used to obtain componentwise upper and lower bounds on the error that is made when the positive eigenvectors of a large nonnegative irreducible matrix have to be approximated by a block decomposition and aggregation technique. The computation of these bounds can itself be regarded as a new approximation technique, called here bounded aggregation. Finally, the particular case of stochastic matrices is analyzed and a numerical example is given.